The present invention relates to the areas of quantum mechanics, quantum computation and fuzzy logic. In particular, the present invention provides a method and system for fuzzy representation of information using quantum mechanics, allowing fuzzy logic operations and fuzzy control operations to be performed via quantum mechanics.
Fuzzy logic is a way of representing and processing information that generalizes ordinary Boolean logic to include a continuous range of propositional values. Applications of classical-mechanically implemented fuzzy logic have ranged from fuzzy logical models of chemical systems to fuzzy control of very large mechanical devices, such as power stations, and distributed systems, such as subways.
Fuzzy information is distinct from digital, binary information in that it is generally represented using the entire range [0,1] of the real numbers, rather than using a binary set of integers such as the set of its endpoints {0,1}.
Fuzziness is usually associated with uncertainty about the nature or state of an object or event. However, fuzziness is distinct from probabilistic uncertainty, though the two are not dissimilar in some respects; for example, they both represent information using the entire numerical range [0,1]. Probability is generally defined as a measure of the frequency with which a random variable takes values inside a specified region in the relevant parameter space, the region representing a xe2x80x9ccrisp set.xe2x80x9d A crisp set is one in which a single event is attributed either of the Boolean values, 0 or 1. For example, in an ideal coin toss, there is a probability associated with the result of the coin landing heads-up. The outcome of a coin toss is either one in which the coin is located in a region of parameter space where the upper side of the coin is xe2x80x9cheads,xe2x80x9d 1 for heads, or not, in which case it must be xe2x80x9ctails,xe2x80x9d 0 for heads. By contrast, with fuzziness the uncertainty of a single event is described by its xe2x80x9cdegree of membershipxe2x80x9d in a region of parameter space, this region representing a xe2x80x9cfuzzy set.xe2x80x9d
For example, the characteristic of age lends itself to fuzzy representation. An evaluation of a person in middle age with regard to xe2x80x9coldnessxe2x80x9d reveals that a middle-age person is to some extent old, and to some extent young. A person is of increasing xe2x80x9coldnessxe2x80x9d with increasing age. For example, a 40-year-old man might have an xe2x80x9coldxe2x80x9d value of 0.4 and xe2x80x9cyoungxe2x80x9d value of 0.6 while a 60-year old might have an xe2x80x9coldxe2x80x9d value of 0.6 and a xe2x80x9cyoungxe2x80x9d value of 0.4. Thus, in the first example, the 40-year old man has a fuzzy membership degree of 0.4 in the fuzzy set xe2x80x9coldxe2x80x9d and a fuzzy membership degree of 0.6 in the fuzzy set xe2x80x9cyoung.xe2x80x9d Similarly, in the second example, the 60-year old man has a fuzzy membership degree of 0.6 in the fuzzy set xe2x80x9coldxe2x80x9d and a fuzzy membership degree of 0.4 in the fuzzy set xe2x80x9cyoung.xe2x80x9d In general, a fuzzy set associates with each object or event a value from the entire interval [0,1], instead of only the set {0,1} as in the case of a crisp set. This value is not taken to represent a frequency, as in the case of a probability. Rather, it is a description of the degree to which the object or event is attributed the property in question.
Fuzzy control is the use of fuzzy information to exert control over an object. A fuzzy rule-set or knowledge base, rather than a traditional modeling algorithm, is used to exert fuzzy control. The xe2x80x9cknowledge basexe2x80x9d is a set of prescribed (or learned) fuzzy rules that first quantifies input data (the xe2x80x9cfuzzificationxe2x80x9d stage), carries out inferences (the fuzzy matching stage), and produces a control output (the xe2x80x9cdefuzzificationxe2x80x9d stage), which is used by an actuator to perform a resulting control action. Devices for carrying out fuzzy control are referred to as xe2x80x9cfuzzy logic controllers.xe2x80x9d
In certain contexts, fuzzy control operates more efficiently than standard control because it does not require: i) the exact solution of the mathematical problem arising from a crisp characterization of the system under control, or ii) highly precise sensing of the system""s state.
The processing of fuzzy information and control operations are traditionally implemented using a computing device such as a general purpose digital computer. Typically, the computing device is equipped with a central processing unit, memory storage, an input/output mechanism and appropriate software to carry out the fuzzy representation and/or control.
FIG. 1a, which is prior art, illustrates a process effected by a fuzzy control system. As shown in FIG. 1a, 102 input received from control object 105 is fuzzified in block 110. Processing block 120 processes fuzzified information, which is then defuzzified at block 130 to generate control output 140. Control output 140 is then used to control control object 105. Note that fuzzify block 110, process block 120 and defuzzify block 130 are typically combined in a single device such as a general purpose digital computer system.
FIG. 1b, which is prior art, illustrates a block diagram of a fuzzy logic control system. Fuzzy logic controller 150 includes processor 160, crisp control actuator 165 and crisp input sensor 170. Fuzzy logic controller 150 is coupled to control object 105 via crisp control actuator 165 and crisp input sensor 170. Crisp input sensor 170 receives input data from control object 105 and transmits this input data to processor 160, which may be, for example, a general-purpose digital computer. Processor 160 performs fuzzy logic processing using input provided by crisp input sensor 170. In particular, processor 160 performs fuzzification, fuzzy logic processing and defuzzification (i.e., blocks 110, 120 and 130 in FIG. 1a) as a function of input provided by crisp input sensor 170. Output generated by processor 160 is used to control actuator 165, which in turn controls control object 105.
FIG. 1c, which is prior art, illustrates a fuzzy control pair. In fuzzy control operations, membership functions corresponding to fuzzy logic propositions form fuzzy control pairs, i.e. fuzzy logic patches or fuzzy rules, known as the xe2x80x9cfuzzy rule set.xe2x80x9d The first proposition of the fuzzy pair (114a-114d) is applied to an input, while the second of the fuzzy pair (117a-117d) is applied to generate an output, which is then utilized for control of a system.
Known methods and systems for performing fuzzy logic operations and fuzzy control such as those depicted in FIG. 1c rely upon conventional/classical computing devices and present inherent limitations for efficient realization of fuzzy logic and control at defuzzification. In particular, typically classical computing architectures are non-parallel, which limits the speed and efficiency in performing fuzzy logic operations. Although significant research has been directed at parallel computing devices, implementing classical fuzzy logic on them still involves a non-negligible number of traditional logical operations.
In recent years, significant research has been directed toward realization of quantum computing devices, which promise significantly greater computational efficiency than conventional classical mechanical digital devices using serial or parallel architectures. Quantum computing is the use of quantum mechanical systems to represent and process information, suggested by Feynman in 1982. Since then, it has been a subject of increasingly active research. The central technology of quantum computing is the quantum computer, a qualitatively novel information processing device.
Aside from the dramatic potential increases in computational efficiency promised by quantum computing devices, the consideration of quantum effects in computation is also necessitated by Moore""s Law. Moore""s Law describes the rate of miniaturization of information processing systems, such as microchips and predicts an inevitable reduction of the size of a digital computer""s functional elements beyond the microscopic realm into the sub-microscopic, quantum realm. Moore""s law strongly suggests that the processing of fuzzy information faces the same inevitability of being carried out at the quantum scale as does purely Boolean information processing.
Conventional approaches for relating quantum mechanics and logic have been built on two and three-valued logics similar to Boolean logic, rather than fuzzy set theory or fuzzy logic (see, for example, Varadarajan, 1985). Likewise, it has recently been suggested that some connection might be made between quantum computing and neural information processing. However, efforts have focused on speculations that the human brain or neural system might be a quantum computer of some sort (for example, as discussed in Penrose""s The Emperor""s New Mind), rather than forging a connection between quantum mechanics and fuzzy logic or fuzzy control.
In addition to general aspects of decreasing scale and optimal computational architecture, classical mechanical implementations of xe2x80x9cdefuzzificationxe2x80x9d are functionally inefficient in that they require an extra step beyond the essential requirements of measurement, because the predictions of classical theory are not necessarily statistical. On the other hand, in quantum mechanics, the form of measurement itself is generally that of performing averages (through repetitions of the event detection process rather than numerically calculating averages), because the predictions of the theory describe the behavior of ensembles. Furthermore, quantum mechanics and fuzzy control have a common basic component, averaging. In fuzzy control, defuzzification is done through the finding of weighted averages. Fuzzy control and quantum mechanics are thus a natural fit. In addition, the uniquely quantum mechanical correlations present in quantum systems can be used to realize the inference stage of fuzzy control, implementing parallel fuzzy rule sets more efficiently and naturally.
Despite the wealth of advantages for using quantum mechanics to perform fuzzy logic, there are no known methods for implementing fuzzy control and fuzzy information processing using quantum systems. Thus, there exists a significant need to design systems for fuzzy information representation and fuzzy control implementation in a quantum system.
The present invention provides a method and system for representing fuzzy information, and performing fuzzy logic and control operations on that information using a quantum system. The representation of fuzzy information using quantum systems and implementation of logic operations and fuzzy control using quantum mechanics provides significant computational and efficiency gains over known methods for performing fuzzy logic and control, which rely on conventional classical computing devices. According to one embodiment, fuzzy logic and control operations are performed using a quantum computer. The present invention extends quantum computing beyond purely Boolean-logical information processing, by providing a quantum mechanical method for representing fuzzy information and realizing fuzzy logic and control.
According to one embodiment, fuzzy set membership functions that define fuzzy propositions are represented via quantum states. This is accomplished using the (projection) probability values associated with these states relative to other states in the same Hilbert space. These values are formed into distributions that serve as fuzzy set membership functions. A very large variety of distributions can be formed from quantum mechanical probability values in this manner, allowing a wide range of fuzzy propositions to be captured utilizing the representation method described herein.
Fuzzy logic operations can then be carried out on this fuzzy information, using quantum mechanical unitary evolutions and measurement processes. This information thus can be used to implement quantum fuzzy logic and to carry out fuzzy control operations quantum mechanically.
The fuzzy information takes the form of distributions of quantum mechanical values (traditionally interpreted as probability values for measurement outcomes) that take the full range [0,1], rather than binary quantum superpositions of states taking Boolean values (that is, xe2x80x9cqubitsxe2x80x9d), as in prior, xe2x80x9ccrispxe2x80x9d forms of quantum information processing.
The invention is a design principle for the representation and processing of fuzzy information in quantum systems for purposes such as logical processing and system control. The principle utilizes quantum probability values to form fuzzy membership functions and specifies quantum mechanical fuzzy logic and fuzzy control operations that use fuzzy set membership functions.